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Weigh up what is gained and lost, when pricing fixed-income derivatives, by shrinking the time steps.

FRM® Part II · Market Risk Measurement and Management · Arbitrage Pricing with Term Structure Models · FRM2.MR.12.i

The idea

Shrinking the time step, say from six months to one month, makes the tree a better approximation of continuous rate movements. It gives more possible rates at each date, finer exercise dates for options, and prices that converge toward the continuous-time model value. The costs are computation and calibration. A recombining tree with n steps has about n²÷2 nodes, so ten times more steps costs roughly one hundred times more work. Parameters must be rescaled with the step. Discounting must use the step's own compounding, for example r÷12 for a monthly step. Write StepVol for the per-step volatility, with σ the annual volatility and dt in years. StepVol = σ × √dt. A smaller step improves the numerics but does not repair a wrong model.

Formulas

Per-step volatility on a rate tree

StepVol = σ × √dt

σ
the annual volatility of the rate
dt
the length of the time step in years

Scale by the square root of the step, not the step itself: dividing an annual volatility by 12 for a monthly step understates it.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which pair correctly states a benefit and a cost of shortening the time step of a binomial interest rate tree, for example from six months to one month?

  1. More realistic rate outcomes and a closer approach to continuous-time prices, at the cost of many more nodes to compute. - correct
  2. Lower computing cost, because each step involves a smaller rate change and so a simpler valuation.Why not: Smaller steps mean more steps and more nodes; the cost rises, it does not fall.
  3. A guarantee that the model is correct, at the cost of needing fewer calibration inputs.Why not: Shorter steps improve the numerics but cannot repair a mis-specified model, and calibration still needs a volatility.
  4. Removal of all discretization effects, at the cost of requiring non-recombining trees.Why not: Discretization error shrinks, not vanishes, and recombining trees remain usable.

Shorter steps give a finer rate distribution and better option values (exercise dates, convergence), while the node count rises roughly with the square of the number of steps.

A harder one

A model assumes the annual volatility of the short rate is 100 bp (absolute, normal). The tree is changed to monthly steps. The per-step standard deviation of the rate change that is consistent with this annual volatility is closest to:

  1. 8.3 bpWhy not: Divides the annual volatility by 12 rather than by the square root of 12.
  2. 346.4 bpWhy not: Multiplies by the square root of 12, which would scale a monthly volatility up to annual.
  3. 28.9 bp - correct
  4. 70.7 bpWhy not: Uses the six-month step (square root of 1/2) rather than the monthly one.

Per-step volatility = sigma x sqrt(dt) = 100 x sqrt(1/12) = 28.9 bp.

The hardest tier

A tree with monthly steps has a one-month rate (annualized, monthly compounding) of 5.00% today, rising to 5.30% or falling to 4.70% after one month with probability 0.5 each. The price of a two-month zero-coupon bond with face value 100 is closest to:

  1. 99.17 - correct
  2. 95.18Why not: Uses semiannual compounding (1 + r/2) for monthly steps instead of 1 + r/12.
  3. 90.70Why not: Uses annual compounding (1 + r) for the monthly steps.
  4. 99.59Why not: Forgets to discount the expected one-month value back one more month at today's 5.00% rate.

Month-1 values: 100/(1 + 0.053/12) = 99.560, 100/(1 + 0.047/12) = 99.610; average 99.585; discount at 5%/12: 99.17. When the step shrinks the discounting must follow the step.

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